Entire function bounded on vertical lines $\Re(z) = \pm1$, but not between them.

Question

Is there an entire function $f : \mathbb{C} \to \mathbb{C}$ such that $f$ is bounded on the vertical lines $\Re(z) = \pm1$, but is unbounded on the region between them i.e. $$ \sup_{|\Re(z)| = 1} |f(z)| < \infty \qquad\text{and}\qquad \sup_{|\Re(z)| < 1} |f(z)| = \infty $$

What approach should be taken to prove/disprove that such a function exists? If it exists can I define it explicitly?

Answer 1

The simplest example is $f(z)=e^{e^{\pi iz/2}}$. Note that for $\Re z =\pm 1, z=\pm 1+iy$ one has $e^{\pi iz/2}=\pm ie^{-\pi y/2}$, so $|f(z)|=1$ there but for $\Re z=0, z=iy$, we have $f(z)=e^{e^{-\pi y/2}}$ which is highly unbounded as $y \to -\infty$

By Phragmen Lindelof, one needs an entire function of infinite order.